A Course in Theoretical Physics

P. John Shepherd

• Publisher: Wiley
• Year: 2013
• Print ISBN: 9781118481424, 9781118481349
• eBook ISBN: 9781118516928
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This book is a comprehensive account of five extended modules covering the key branches of twentieth-century theoretical physics, taught by the author over a period of three decades to students on bachelor and master university degree courses in both physics and theoretical physics.

The modules cover nonrelativistic quantum mechanics, thermal and statistical physics, many-body theory, classical field theory (including special relativity and electromagnetism), and, finally, relativistic quantum mechanics and gauge theories of quark and lepton interactions, all presented in a single, self-contained volume.

In a number of universities, much of the material covered (for example, on Einstein's general theory of relativity, on the BCS theory of superconductivity, and on the Standard Model, including the theory underlying the prediction of the Higgs boson) is taught in postgraduate courses to beginning PhD students.

A distinctive feature of the book is that full, step-by-step mathematical proofs of all essential results are given, enabling a student who has completed a high-school mathematics course and the first year of a university physics degree course to understand and appreciate the derivations of very many of the most important results of twentieth-century theoretical physics.

Module III

Many-Body Theory

13

Quantum Mechanics and Low–Temperature Thermodynamics of Many–Particle Systems

13.1 Introduction

By a “many–particle system” we shall mean a system of macroscopic size (for example, one for which the number of particles is of the order of 1023).

By “low–temperature” we shall mean that the temperature T is such that (a) the system is only weakly excited relative to the ground state and (b) quantum statistics is necessary (Fermi–Dirac for fermions, Bose–Einstein for bosons), that is, the quantum–mechanical principle of indistinguishability of identical particles has macroscopic consequences. (An example of the latter is the dramatic difference in the properties of liquid helium–3 and liquid helium–4.)

The properties to be studied in this chapter include:

(a) the wave function and energy of the ground state of the system;

(b) the wave functions, energies and widths (inverse lifetimes) of weakly excited states;

(c) the thermodynamic state variables [found from the energies from (b) by the methods of statistical mechanics].

13.2 Systems of noninteracting particles

The hamiltonian

of a system of noninteracting particles can be written as:

(13.1)

where

n is the hamiltonian of particle n, and

clearly contains no potential energies of interaction of the type

nm.

The single–particle energy eigenstates (eigenstates of

n) may be written as

(13.2)

with corresponding energy eigenvalues εαi:

(13.3)

The symbol α denotes the set of quantum numbers corresponding to a complete set of single–particle observables. For example, for an electron moving in the Coulomb potential of a nucleus the set α could be the set n,l,m,ms (see section 1.17 on hydrogenic states), while for a free particle with spin the set α could be the set kx, ky, kz, ms. The different sets of possible numerical values of the quantum numbers in a given set α are labelled by the index i, which may be considered to take integer values from 1 to ∞, using any chosen ordering convention. For example, we could choose i = 1 to correspond to the set of quantum-number values for the ground state of the part...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• Notation
• Preface
• Module I: Nonrelativistic Quantum Mechanics
• Module II: Thermal and Statistical Physics
• Module III: Many-Body Theory
• Module IV: Classical Field Theory and Relativity
• Module V: Relativistic Quantum Mechanics and Gauge Theories
• Appendices
• Supplementary Reading
• Index